In finite-dimensional spaces the sum range of a series has to be an affine subspace. It has long been known that this is not the case in infinite-dimensional Banach spaces. In particular in 1984 M. I. Kadets and K. Woźniakowski obtained an example of a series whose sum range consisted of two points, and asked whether it was possible to obtain more than two, but finitely many points. This paper answers this question affirmatively, by showing how to obtain an arbitrary finite set as the sum range of a series in any infinite-dimensional Banach space.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-4,
author = {Jakub Onufry Wojtaszczyk},
title = {A series whose sum range is an arbitrary finite set},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {261-281},
zbl = {1089.46010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-4}
}
Jakub Onufry Wojtaszczyk. A series whose sum range is an arbitrary finite set. Studia Mathematica, Tome 166 (2005) pp. 261-281. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-4/